[wu::riddles] Geometric In Arithmetic

Today we share our analysis of the following problem, which kindly comes to us from Colombian engineer Ariel Nunez, a former Iberoamericano Math Olympiad Bronze medalist!

Problem:

Given a real arithmetic progression , some terms are deleted to produce a geometric sequence with ratio . Find all possible values of .

Our Solution:

Define for and . The geometric subsequence is for , where is the indexing sequence. Evaluating ,

Let . From the above recurrence, for some constant . From the initial condition ,, which implies . Equating these expressions for ,

Since , each imposes a divisibility relationship between and .
From the case of , , so for some . Consequently,

and must both be positive for to be non-negative for all . By induction since , it follows that

This constraint cannot be satisfied if is irrational. Suppose , such that for non-commensurate . From the case of , implies that for some not divisible by . However, from the case of , can be rewritten as , which implies that is divisible by — which is a contradiction. Therefore, can only be a natural number. is achievable when , and all other natural numbers for are achievable when .